https://file.aiquickdraw.com/mathbot/file/1727309537972p5zqz5k6.jpg

The steps shown in your image are essentially correct, but let's break it down and analyze the details.

You start with the expression:
\[
|2x - 3|
\]
This represents the **absolute value**, which can be interpreted as the distance between \( 2x \) and \( 3 \) on a number line.

### Steps Explanation:

1. **Expression \( |2x - 3| \)**:
   This is interpreted as the absolute distance between \( 2x \) and \( 3 \). So, the original expression is correct in describing the "distance between \( 2x \) and 3."

2. **Factoring out \( -1 \)**:
   The next step is factoring out the \( -1 \) from inside the absolute value expression:
   \[
   |-1(-2x + 3)|
   \]
   This is mathematically sound because pulling out a negative factor doesn’t change the overall value inside the absolute value. The absolute value function ignores signs. 

3. **Simplification \( |-2x + 3| \)**:
   You remove the \( -1 \) because:
   \[
   |-1 \cdot (-2x + 3)| = | -2x + 3 |
   \]
   So this is also correct. The absolute value of a negative or positive expression is the same as that of the original expression.

4. **Conclusion**:
   The expression \( |-2x + 3| \) represents the distance between \( -2x \) and 3.

### Final Verdict: 
Your steps are correct, and the final result shows that \( |2x - 3| \) is equivalent to \( |-2x + 3| \), as both represent the same distance.

Do you want details on how absolute values represent distances or any clarifications? Here are five questions to help expand your understanding:

1. What is the geometric interpretation of absolute value on a number line?
2. How does factoring affect expressions inside an absolute value?
3. Why is \( |a| = |-a| \) true for any number \( a \)?
4. Can we generalize the property that \( |f(x)| = |-f(x)| \) for any linear function \( f(x) \)?
5. What happens if we add or subtract constants from absolute value expressions?

**Tip**: Remember, the absolute value of any real number is its distance from 0, regardless of direction on the number line!

What is the interpretation of the expression |2x - 3|, and how does it relate to the distance between points?

Understand how the absolute value expression |2x - 3| represents the distance between 2x and 3. This explanation covers absolute value properties, factoring, and interpreting distance on a number line. Ideal for students in Grades 7-10.

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